, Volume 86, Issue 1, pp 131-159

Cyclic homology and the Macdonald conjectures

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LetA+(k) denote the ring ℂ[t]/t k+1 and letG be a reductive complex Lie algebra with exponentsm 1, ...,m n. This paper concerns the Lie algebra cohomology ofGA +(k) considered as a bigraded algebra (here one of the gradings is homological degree and the other, which we callweight, is inherited from the obvious grading ofGA +(k)). We conjecture that this Lie algebra cohomology is an exterior algebra withk+1 generators of homological degree 2m s +1 fors=1,2, ...,n. Of thesek+1 generators of degree 2m s +1, one has weight 0 and the others have weights (k+1)m s +t fort=1,2, ...,k.

It is shown that this conjecture about the Lie algebra cohomology of ⊗A +(k) implies the Macdonald root system conjectures. Next we consider the case thatG is a classical Lie algebra with root systemA n ,B n ,C n , orD n. It is shown that our conjecture holds in the limit onn asn approaches infinity which amounts to the computation of the cyclic and dihedral cohomologies ofA+(k). Lastly we discuss the relevance of this limiting case to the case of finiten in this situation.